A simple illustration of the Mean Value theorem (particularly good for a less theoretical course, but I like it in any setting): A man is photographed at a tollbooth at 12:00, and then arrives another tollbooth, 250 miles down the road, at 2:00. A cop pulls him over and gives him a traffic ticket for driving 125 mph.
His defense lawyer claims in court, "You can't prove that there was ever any particular moment when my client was actually driving 125 mph..."
